SNUB DODECAHEDRON

The 'snub dodecahedron', or 'snub icosidodecahedron', is an Archimedean solid.
The snub dodecahedron has 92 faces, of which 12 are pentagons and the other 80 are equilateral triangles. It also has 150 edges, and 60 vertices. It has two distinct forms, which are mirror images (or "enantiomorphs") of each other.

Contents

Geometric relations

Cartesian coordinates

See also

References

External links

Geometric relations

The snub dodecahedron can be generated by taking the twelve pentagonal faces of the dodecahedron, pulling them outward so they no longer touch. This creates the rhombicosidodecahedron. Then give them all a small rotation on their centers (all clockwise or all counter-clockwise) until the space between can be filled by equilateral triangles.

Dodecahedron

Rhombicosidodecahedron (''Expanded dodecahedron'')

Archimedes, an ancient Greek who showed major interest in polyhedral shapes wrote a treatise on thirteen semi-regular solids. Snub-Dodecahedron belongs to the thirteen semi-regular solids.

Cartesian coordinates

Cartesian coordinates for the vertices of a snub dodecahedron are all the even permutations of
: (±2α, ±2, ±2β),
: (±(α+β/τ+τ), ±(-ατ+β+1/τ), ±(α/τ+βτ-1)),
: (±(-α/τ+βτ+1), ±(-α+β/τ-τ), ±(ατ+β-1/τ)),
: (±(-α/τ+βτ-1), ±(α-β/τ-τ), ±(ατ+β+1/τ)) and
: (±(α+β/τ-τ), ±(ατ-β+1/τ), ±(α/τ+βτ+1)),
with an even number of plus signs, where
: α = ξ-1/ξ
and
: β = ξτ+τ²+τ/ξ,
where τ = (1+√5)/2 is the golden mean and
ξ is the real solution to ξ³-2ξ=τ, which is the beautiful number
:
or approximately 1.7155615.
Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.

See also

★ and spinning snub dodecahedron

★ Snub cube

★ Snub hexagonal tiling

References

★ The Geometrical Foundation of Natural Structure: A Source Book of Design, , Robert, Williams, Dover Publications, Inc, 1979, ISBN 0-486-23729-X (Section 3-9)

External links

★

★ The Uniform Polyhedra

★ Virtual Reality Polyhedra The Encyclopedia of Polyhedra

★ Irregular Snub Dodecahdron Wire Model Math Models, Vincent Herr